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Differential Equations: A Modeling Perspective, 2nd Edition

Differential Equations: A Modeling Perspective, 2nd Edition

Robert L. Borrelli, Courtney S. Coleman

ISBN: 978-0-471-43332-3

Jan 2004

736 pages

Select type: Hardcover

Out of stock

$248.95

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This effective and practical new edition continues to focus on differential equations as a powerful tool in constructing mathematical models for the physical world.  It emphasizes modeling and visualization of solutions throughout. Each chapter introduces a model and then goes on to look at solutions of the differential equations involved using an integrated analytical, numerical, and qualitative approach.  The authors present the material in a way that’s clear and understandable to students at all levels.  Throughout the text the authors convey their enthusiasm and excitement for the study of ODEs.

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This item: Differential Equations: A Modeling Perspective, 2nd Edition

Differential Equations: A Modeling Perspective, 2nd Edition

(Hardcover $248.95)

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(Paperback $89.95)

Original Price:$338.90

Purchased Together:$318.95

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1: Modeling and Differential Equations.

1.1 The Modeling Approach.

1.2 A Modeling Adventure.

1.3 Models and Initial Value Problems.

1.4 The Modeling Process: Differential Systems.

SPOTLIGHT ON MODELING: RADIOCARBON DATING.

SPOTLIGHT ON MODELING: COLD MEDICATION I.

2: First-Order Differential Equations.

2.1 Linear Differential Equations.

2.2 Linear Differential Equations: Qualitative Analysis.

2.3 Existence and Uniqueness of Solutions.

2.4 Visualizing Solution Curves: Slope Fields.

2.5 Separable Differential Equations: Planar Systems.

2.6 A Predator-Prey Model: the Lotka-Volterra System.

2.7 Extension of Solutions: Long-Term Behavior.

2.8 Qualitative Analysis: State Lines, Sign Analysis.

2.9 Bifucations: A Harvested Logistic Model.

Snapshot on Solution Formula Techniques.

SPOTLIGHT ON APPROXIMATE NUMERICAL SOLUTIONS.

SPOTLIGHT ON COMPUTER IMPLEMENTATION.

SPOTLIGHT ON STEADY STATES: LINEAR ODES.

SPOTLIGHT ON MODELING: COLD MEDICATION II.

SPOTLIGHT ON CHANGE OF VARIABLES: PURSUIT MODELS.

SPOTLIGHT ON CONTINUITY IN THE DATA.

3: Second-Order Differential Equations.

3.1 Models of Springs.

3.2 Undriven Constant-Coefficient Linear Differential Equations.

3.3 Visualizing Graphs of Solutions: Direction Fields.

3.4 Periodic Solutions: Simple Harmonic Motion.

3.5 Driven Linear ODEs: Undetermined Coefficients I.

3.6 Driven Linear ODEs: Undetermined Coefficients II.

3.7 Theory of Second-Order Linear Differential Equations.

3.8 Nonlinear Second-Order Differential Equations.

A Snapshot Look at Constant-Coefficient Polynomial Operators.

SPOTLIGHT ON MODELING: VERTICAL MOTION.

SPOTLIGHT ON MODELING: SHOCK ABSORBERS.

SPOTLIGHT ON EINSTEIN'S FIELD EQUATIONS.

4: Applications of Second-Order Differential Equations.

4.1 Newton's Laws: The Pendulum.

4.2 Beats and Resonance.

4.3 Frequency Response Modeling.

4.4 Electrical Circuits.

Snapshot on Mechanical and Electrical Models.

SPOTLIGHT ON MODELING: TUNING A CIRCUIT.

5: The Laplace Transform.

5.1 The Laplace Transform: Solving IVPs.

5.2 Working with the Transform.

5.3 Transforms of Periodic Functions.

5.4 Convolution.

SPOTLIGHT ON THE DELTA FUNCTION.

SPOTLIGHT ON MODELING: TIME DELAYS AND COLLISIONS.

6: Linear Systems of Differential Equations.

6.1 Compartment Models: Tracking Lead.

6.2 Eigenvalues, Eigenvectors and Eigenspaces of Matrices.

6.3 Undriven Linear Differential Systems: Real Eigenvalues.

6.4 Undriven Linear Systems: Complex Eigenvalues.

6.5 Orbital Portraits for Planar Systems.

6.6 Driven Systems: The Matrix Exponential.

6.7 Steady States.

6.8 The Theory of General Linear Systems.

SPOTLIGHT ON VECTORS, MATRICES, INDEPENDENCE.

SPOTLIGHT ON LINEAR ALGEBRAIC EQUATIONS.

SPOTLIGHT ON BIFURCATIONS: SENSITIVITY.

7: Nonlinear Differential Systems.

7.1 Chemical Kinetics: The Fundamental Theorem.

7.2 Properties of Autonomous Systems, Direction Fields.

7.3 Interacting Species: Cooperation, Competition.

SPOTLIGHT ON MODELING: DESTRUCTIVE COMPETITION.

SPOTLIGHT ON MODELING: BIFURCATION AND SENSITIVITY.

8: Stability.

8.1 Stability and Linear Autonomous Systems.

8.2 Stability and Nonlinear Autonomous Systems.

Stability of PlanarNonlinear Systems.

8.3 Conservative Systems.

SPOTLIGHT ON LYAPUNOV FUNCTIONS.

SPOTLIGHT ON ROTATING BODIES.

9: Nonlinear Systems: Cycles and Chaos.

9.1 Cycles.

9.2 Solution Behavior in Planar Autonomous Systems.

9.3 Bifucations.

9.4 Chaos.

SPOTLIGHT ON CHAOTIC SYSTEMS.

10: Fourier Series and Partial Differential Equations.

10.1 Vibrations of a Guitar String.

10.2 Fourier Trigonometric Series.

10.3 Half-Range and Exponential Fourier Series.

10.4 Temperature in a Thin Rod.

10.5 Sturm-Liouville Problems.

10.6 The Method of Eigenfunction Expansions.

SPOTLIGHT ON DECAY ESTIMATES.

SPOTLIGHT ON THE OPTIMAL DEPTH FOR A WINE CELLAR.

SPOTLIGHT ON APPROXIMATION OF FUNCTIONS.

11: Series Solutions.

11.1 The Method of Power Series.

11.2 Series Solutions Near an Ordinary Point.

11.3 Regular Singular Points: Euler's ODE.

11.4 Series Solutions Near Regular Singular Points.

SPOTLIGHT ON LEGENDRE POLYNOMIALS.

SPOTLIGHT ON BESSEL FUNCTIONS.

Appendix A: Basic Theory of Initial Value Problems.

A.1 Uniqueness.

A.2 The Picard Process for Solving an Initial Value Problem.

A.3 Extention of Solutions.

Appendix B: Background Information.

B.1 Power Series.

B.2 Results from Calculus.

Answers to Selected Problems.

Index.

Web Spotlights.

*27 Additional SPOTLIGHTS appear on the text's Web site at www.wiley.com/college/borrelli.

  • Emphasizes mathematical modeling every picture tells a story.
  • A dynamical systems approach that models natural processes that evolve in time.
  • Graphic visualization is emphasized - The text and the hundreds of graphs and solutions emphasize a visual connection with the theory.